Shelah's excellence conjecture for atomic classes

Let TT be a countable first-order theory and let KTK_T be its class of atomic models. Excellence conjecture. If KTK_T is categorical in every n\beth_n for n<0n<\beth_0, then KTK_T is excellent. The claim would show that the counterexamples discussed in the source are best possible; the source says this appears far away and treats it as open.

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Primary source

Saharon Shelah, “A.E.C. with not too many models”, arXiv:1302.4841 (2013).

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